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Math Help - Jordan blocks

  1. #1
    Senior Member bkarpuz's Avatar
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    Jordan blocks

    Dear MHF members,

    my problem reads as follows. I have some ideas about some parts of the problem but I would like to share it completely because I find the problem nice.
    Problem. Consider the system \mathbf{X}^{\prime}=\mathbf{A}\mathbf{X}, where \mathbf{A} is a 30\times30 constant matrix with only two distinct eigenvalues \lambda_{1}=1 and \lambda_{2}=2 Suppose that the Jordan form \mathbf{J} of \mathbf{A} has Jordan blocks \mathbf{J}_{3},\mathbf{J}_{3},\mathbf{J}_{6} corresponding to \lambda_{1} and blocks \mathbf{J}_{1},\mathbf{J}_{2},\mathbf{J}_{3}, \mathbf{J}_{3},\mathbf{J}_{4},\mathbf{J}_{5} corresponding to \lambda_{2}, where \mathbf{J}_{k} is a k\times k Jordan block.

    1. How many linearly independent eigenvectors does \mathbf{A} have?
    2. Suppose that \mathbf{V} is not an eigenvector but satisfies (\mathbf{A}-\lambda_{2}\mathbf{I})^{3}\mathbf{V}=0. How many linearly independent solution vectors \mathbf{V} are there?
    3. Suppose that the Jordan blocks occur down the diagonal of the \mathbf{J} in the order stated above. Which matrix elements of \lim_{t\to\infty}t^{4}\mathrm{e}^{-\lambda_{2}t}\mathrm{e}^{\mathbf{J}t} are non-zero and what are their values? \rule{0.2cm}{0.2cm}

    Thanks a lot.
    bkarpuz
    Last edited by bkarpuz; May 11th 2011 at 10:49 AM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint for 1


    \dim \ker (J_k-\lambda I_k)=k-\textrm{rank}(J_k-\lambda I_k)=k-(k-1)=1


    Quote Originally Posted by bkarpuz View Post
    I have some ideas about some parts of the problem but I would like to share it completely because I find the problem nice.

    Can you comment those ideas?
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  3. #3
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by FernandoRevilla View Post
    Can you comment those ideas?
    1. The number of regular eigenvectors are 9, and the remaining 21 are generalized eigenvectors.
    2. I feel like it is 15, but I really need help with this one.
    3. The terms of the blocks of the exponential matrix corresponding to \lambda_{1}=1 will all vanish. For \lambda_{2}=2, all the terms except one will vanish. The non-zero term is located at (26,30), and its value is 1/4!.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by bkarpuz View Post
    1. The number of regular eigenvectors are 9, and the remaining 21 are generalized eigenvectors.

    Right.


    2. I feel like it is 15, but I really need help with this one.

    Right. Denote J_{k,\lambda_i} a Jordan block of order k associated to the eigenvalue \lambda_i. Then,

    (i)\quad \textrm{rank} \begin{bmatrix}{J_{3,\lambda_1}-\lambda_2 I_3}&{0}&{0}\\{0}&{{J_{3,\lambda_1}-\lambda_2 I_3}&{0}\\{0}&{0}&{{J_{6,\lambda_1}-\lambda_2 I_6}\end{bmatrix}^3=3+3+6=12


    (ii)\quad \textrm{rank}(J_{k,\lambda_2}-\lambda_2I_k)^3=\begin{Bmatrix}{ 0}&\mbox{ if }& k=1,2,3\\1 & \mbox{if}& k=4\\2 & \mbox{if}& k=5\end{matrix}

    According to the canonical form structure, we verify

    \dim (\ker (A-\lambda_2I_{30})^3)=30-\textrm{rank}(A-\lambda_2I_{30})^3=30-15=15


    3. The terms of the blocks of the exponential matrix corresponding to \lambda_{1}=1 will all vanish. For \lambda_{2}=2, all the terms except one will vanish. The non-zero term is located at (26,30), and its value is 1/4!.

    Right. The non zero term corresponds to the block

    J_{5,\lambda_2}
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